Comparing proton momentum distributions in A
= 2 and 3 nuclei via [superscript 2]H [superscript
3]H and [superscript 3]He (e,e#p) measurements
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Cruz-Torres, R. et al. "Comparing proton momentum distributions in
A = 2 and 3 nuclei via 2H 3H and 3He (e, e'p) measurements" Physics
Letter B 797 (August 2019): 134890 © 2019 The Author(s)
As Published
10.1016/J.PHYSLETB.2019.134890
Publisher
Elsevier BV
Version
Final published version
Citable link
https://hdl.handle.net/1721.1/129743
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Creative Commons Attribution 4.0 International license
Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Comparing
proton
momentum
distributions
in
A
=
2 and
3 nuclei
via
2
H
3
H
and
3
He
(
e
,
e
p
)
measurements
Jefferson
Lab
Hall
A
Tritium
Collaboration
R. Cruz-Torres
a,
S. Li
b,
F. Hauenstein
c,
A. Schmidt
a,
D. Nguyen
d,
D. Abrams
d,
H. Albataineh
e,
S. Alsalmi
f,
D. Androic
g,
K. Aniol
h,
W. Armstrong
i,
J. Arrington
i,
H. Atac
j,
T. Averett
k,
C. Ayerbe Gayoso
k,
X. Bai
d,
J. Bane
l,
S. Barcus
k,
A. Beck
a,
V. Bellini
m,
H. Bhatt
n,
D. Bhetuwal
n,
D. Biswas
o,
D. Blyth
i,
W. Boeglin
p,
D. Bulumulla
c,
A. Camsonne
q,
J. Castellanos
p,
J.-P. Chen
q,
E.O. Cohen
r,
S. Covrig
q,
K. Craycraft
l,
B. Dongwi
o,
M. Duer
r,
B. Duran
j,
D. Dutta
n,
E. Fuchey
s,
C. Gal
d,
T.N. Gautam
o,
S. Gilad
a,
K. Gnanvo
d,
T. Gogami
t,
J. Gomez
q,
C. Gu
d,
A. Habarakada
o,
T. Hague
f,
O. Hansen
q,
M. Hattawy
i,
O. Hen
a,
∗
,
D.W. Higinbotham
q,
E. Hughes
u,
C. Hyde
c,
H. Ibrahim
v,
S. Jian
d,
S. Joosten
j,
A. Karki
n,
B. Karki
w,
A.T. Katramatou
f,
C. Keppel
q,
M. Khachatryan
c,
V. Khachatryan
x,
A. Khanal
p,
D. King
y,
P. King
w,
I. Korover
z,
T. Kutz
x,
N. Lashley-Colthirst
o,
G. Laskaris
a,
W. Li
aa,
H. Liu
u,
N. Liyanage
d,
D. Lonardoni
ab,
ac,
R. Machleidt
ad,
L.E. Marcucci
ae,
af,
P. Markowitz
p,
R.E. McClellan
q,
D. Meekins
q,
S. Mey-Tal Beck
a,
Z.-E. Meziani
j,
R. Michaels
q,
M. Mihoviloviˇc
ag,
ah,
ai,
V. Nelyubin
d,
N. Nuruzzaman
o,
M. Nycz
f,
R. Obrecht
s,
M. Olson
aj,
L. Ou
a,
V. Owen
k,
B. Pandey
o,
V. Pandey
ak,
A. Papadopoulou
a,
S. Park
x,
M. Patsyuk
a,
S. Paul
k,
G.G. Petratos
f,
E. Piasetzky
r,
R. Pomatsalyuk
al,
S. Premathilake
d,
A.J.R. Puckett
s,
V. Punjabi
am,
R. Ransome
an,
M.N.H. Rashad
c,
P.E. Reimer
i,
S. Riordan
i,
J. Roche
w,
F. Sammarruca
ad,
N. Santiesteban
b,
B. Sawatzky
q,
E.P. Segarra
a,
B. Schmookler
a,
A. Shahinyan
ao,
S. Širca
ag,
ah,
N. Sparveris
ap,
T. Su
f,
R. Suleiman
q,
H. Szumila-Vance
q,
A.S. Tadepalli
an,
L. Tang
q,
W. Tireman
aq,
F. Tortorici
m,
G. Urciuoli
ar,
M. Viviani
af,
L.B. Weinstein
c,
B. Wojtsekhowski
q,
S. Wood
q,
Z.H. Ye
i,
Z.Y. Ye
as,
J. Zhang
xaMassachusetts Institute of Technology, Cambridge, MA, USA bUniversity of New Hampshire, Durham, NH, USA cOld Dominion University, Norfolk, VA, USA dUniversity of Virginia, Charlottesville, VA, USA eTexas A & M University, Kingsville, TX, USA fKent State University, Kent, OH, USA gUniversity of Zagreb, Zagreb, Croatia hCalifornia State University, Los Angeles, CA, USA
iPhysics Division, Argonne National Laboratory, Lemont, IL, USA jTemple University, Philadelphia, PA, USA
kThe College of William and Mary, Williamsburg, VA, USA lUniversity of Tennessee, Knoxville, TN, USA
mINFN Sezione di Catania, Italy
nMississippi State University, Miss. State, MS, USA oHampton University, Hampton, VA, USA pFlorida International University, Miami, FL, USA qJefferson Lab, Newport News, VA, USA
rSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
*
Correspondingauthor.E-mail address:hen@mit.edu(O. Hen). https://doi.org/10.1016/j.physletb.2019.134890
0370-2693/©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
2 Jefferson Lab Hall A Tritium Collaboration / Physics Letters B 797 (2019) 134890
sUniversity of Connecticut, Storrs, CT, USA tTohoku University, Sendai, Japan uColumbia University, New York, NY, USA vCairo University, Cairo, Egypt wOhio University, Athens, OH, USA
xStony Brook University, State University of New York, NY, USA ySyracuse University, Syracuse, NY, USA
zNuclear Research Center -Negev, Beer-Sheva, Israel aaUniversity of Regina, Regina, SK, Canada
abFacility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA acTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA adDepartment of Physics, University of Idaho, Moscow, ID 83844, USA
aeDepartment of Physics “E. Fermi”, University of Pisa, Italy afINFN, Pisa, Italy
agUniversity of Ljubljana, Ljubljana, Slovenia
ahFaculty of Mathematics and Physics, Jožef Stefan Institute, Ljubljana, Slovenia
aiInstitut für Kernphysik, Johannes Gutenberg-Universität Mainz, DE-55128 Mainz, Germany ajSaint Norbert College, De Pere, WI, USA
akCenter for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA alInstitute of Physics and Technology, Kharkov,
Ukraine
amNorfolk State University, Norfolk, VA, USA anRutgers University, New Brunswick, NJ, USA aoYerevan Physics Institute, Yerevan, Armenia apTel Aviv University, Tel Aviv, Israel
aqNorthern Michigan University, Marquette, MI, USA arINFN, Rome, Italy
asUniversity of Illinois-Chicago, IL, USA
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Article history:
Received11June2019
Receivedinrevisedform14August2019 Accepted21August2019
Availableonline28August2019 Editor:D.F.Geesaman
We report the first measurement of the (e,ep) reaction cross-section ratios for Helium-3 (3He),
Tritium (3H), and Deuterium (d). The measurement covered a missing momentum range of 40≤
pmiss≤550 MeV/c,atlarge momentumtransfer (Q2≈1.9 (GeV/c)2) and xB>1,whichminimized
contributions from non quasi-elastic (QE) reaction mechanisms. The data is compared with
plane-wave impulse approximation (PWIA) calculations using realistic spectral functions and momentum
distributions.ThemeasuredandPWIA-calculatedcross-sectionratiosfor3He/d and3H/d extendtojust
above the typicalnucleonFermi-momentum(kF ≈250 MeV/c) and differfrom eachotherby ∼20%,
whilefor3He/3H theyagreewithinthemeasurementaccuracyofabout3%.Atmomentaabovek
F,the measured3He/3H ratiosdifferfromthecalculationby20%−50%.Finalstateinteraction(FSI)calculations usingthe generalizedEikonalApproximationindicatethat FSIshould changethe3He/3H cross-section
ratio for this measurementby less than5%. Ifthese calculationsare correct, thenthe differences at large missing momentabetween the 3He/3H experimental and calculated ratios could be dueto the
underlyingN N interaction,andthuscouldprovidenewconstraintsonthepreviouslyloosely-constrained short-distancepartsoftheN N interaction.
©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
Nuclearinteractionmodelsareacrucialstartingpointfor mod-ern calculationsofnuclear structureandreactions,aswell asthe properties of dense astrophysical objects such as neutron stars. Phenomenologicalormeson-theoretictwo-bodypotentials,suchas Argonne-V18 (AV18)and CD-Bonn,were developed inthe 1990s usingconstraintsprimarily fromnucleon-nucleon (N N)scattering data [1,2].Morerecently,chiraleffectivefieldtheory(EFT)hasled to the development ofpotentials with systematicand controlled approximations [3,4]. Light atomic nuclei have played a crucial roleinconstraining modern nuclearinteractionmodels, including many-bodyforces,asmanyoftheirproperties(e.g.,charge distri-butionsandradii,ground- andexcited-stateenergies)canbeboth precisely measured and exactly calculated for a given two- and three-nucleoninteractionmodel [5–10].
Whilethe combinationof N N scattering andlight-nucleidata allows one to constrain the two- and three-nucleon interaction at large distances, its short-ranged behavior is still largely un-constrained. The latter is important for understanding nucleon-nucleonshort-range correlations(SRC) in nuclei[11,12],their re-lation to the partonic structure of bound nucleons [13–17], and thestructureofneutronstars [18,19].
Constraining the short-ranged part of the nuclear interac-tion requires studying nucleon momentum distributions at high-momentum. However, previous attempts to extract these were largely unsuccessful, due to the fact that nucleon momentum distributions are not direct observables, and typical experimen-talextractionssufferfromlargereactionmechanismeffects.These introduce significant model-dependent correctionsthat mask the underlying characteristics of the momentum distribution, espe-ciallyathigh-momentum [20–23].
Advances in nuclear reactiontheory now allow usto identify observableswithincreasedsensitivitytonucleonmomentum den-sities at high-momentum [18,24–27]. In light of these advances, we report on a newstudyof themomentum distribution of nu-cleons in Helium-3 relative to Tritium over a broad momentum range.
WestudynucleonmomentumdistributionsusingQuasi-Elastic (QE) electron scattering. In these experiments, an electron with momentum
pe isscatteredfrom thenucleus, transferring energy
ω
andmomentumq to the nucleus. We choose
ω
and q tobe appropriate for elastic scattering from a moving bound nucleon. Bydetectingtheknocked-outproton(
pp)incoincidence withthe
scatteredelectron (
pe), we can measure the missing energy and missingmomentumofthereaction:
Emiss
=
ω
−
Tp−
TA−1,
(1)pmiss
=
pp−
q,
(2)whereq
=
pe−
pe isthemomentumtransfer, TA−1= (
ω
+
mA−
Ep
)
−
(
ω
+
mA−
Ep)
2− |
pmiss|
2 isthereconstructedkineticen-ergyoftheresidual A
−
1 system,andTpandEp arethemeasuredkineticandtotalenergiesoftheoutgoingproton.
InthePlane-WaveImpulse Approximation(PWIA)forQE scat-tering,where asingle exchanged photon isabsorbed on asingle protonandtheknocked-outprotondoesnotre-interactasitleaves thenucleus, thecross-sectionfor A
(
e,
ep)
, electron-induced pro-tonknockoutfromnucleusA,canbewrittenas [28,29]:d6
σ
d
ω
dEpded
p
=
K
σ
epS(
|
pi|,Ei)
(3)where
σep
is the cross-section for scattering an electron from a bound proton [29], K=
Ep|
pp|
is a kinematicalfactor, de and
d
p are the electron and proton solid angles respectively, and
S
(
|
pi|,
Ei)
is the spectral function, which definesthe probabilitytofindaprotoninthe nucleuswithmomentum
|
pi|
andsepara-tion energy Ei.The nucleon momentum distribution is the
inte-gralofthe spectralfunctionover theseparationenergy:n
(
|
pi|)
=
S
(
|
pi|,
Ei)
dEi.InPWIA,the missingmomentum andenergyequalthe initial momentum and separation energy of the knocked-out nucleon:
pi
=
pmiss, Ei=
Emiss.However, thereare other,non-QE, reactionmechanisms,includingfinal state interactions(the rescatteringof theknocked-outproton,FSI),meson-exchangecurrents(MEC),and excitingisobarconfigurations(IC)that canleadtothesame mea-suredfinalstate. Thesealsocontribute to thecrosssection, com-plicatingthissimplepicture.Inaddition,relativisticeffectscanbe significant[30–32].
Previousmeasurementsofthe3He
(
e,
ep)
two- andthree-bodybreakupcross-sectionsweredoneat Q2
=
1.
5 (GeV/c)2 andxB≡
Q22mpω
=
1 wheremp istheprotonmass [21,22],neartheexpectedmaximumoftheprotonrescattering.Themeasured cross-sections disagreed by up to a factor of five with PWIA calculations for
pmiss
>
250 MeV/c. These deviationswere described to goodac-curacybycalculationswhichincludedthecontributionofnon-QE reactionmechanisms,primarilyFSI [18,24–26].Thelarge contribu-tionofsuchnon-QEreactionmechanismstothemeasured
(
e,
ep)
cross-sectionslimited their ability to constrain the nucleon mo-mentumdistributionathighmomenta.
Guidedby reactionmechanismcalculations, which agreewith previous measurements, we can reduce the effect of FSI in two ways [25,27,33–37] by: (A) constraining the angle between
precoil
= −
pmiss and q tobe
θ
rq40◦ and (B) taking the ratioof
(
e,
ep)
cross-sections for same-mass nuclei. The effect of FSI should be similar inboth nucleibecause knocked-outprotons in bothnucleicanrescatter fromthesamenumberofnucleonsand FSIshouldthereforelargelycancelintheratio.Additional non-QE reaction mechanisms such as MEC and IC were shown to be suppressed for Q2
≡
q2−
ω
2>
1.
5 (GeV/c)2and xB
>
1 [33,38]. Thus, the ratio of 3He(
e,
ep)
to 3H(
e,
ep)
cross-sectionsinQE kinematicsat Q2
>
1.
5 (GeV/c)2, xB>
1 andθ
rq40◦ should have increased sensitivity to the ratio of theirspectralfunctions.
Wemeasuredtheratiosofd,3He,and3H
(
e,
ep)
cross-sections in Hall A of the Thomas Jefferson National Accelerator Facility (JLab) using the two high-resolution spectrometers (HRS) and a20 μA 4.326 GeV electron beam incident on one of four 25-cm longgastarget cells [39]. Thefouridenticalcellswere filledwith Hydrogen(70
.
8±
0.
4 mg/cm2), Deuterium(142.
2±
0.
8 mg/cm2), 3He (53.
4±
0.
6 mg/cm2)andTritium(85.
1±
0.
8 mg/cm2)gas [40].We detected the scattered electrons in the left HRS at a cen-tral angle
θ
e=
20.
88◦ and momentum pe=
3.
543 GeV/c,corre-spondingtoacentralfour-momentumtransfer Q2
=
2.
0 (GeV/c)2,energy transfer
ω
=
0.
78 GeV, and xB=
1.
4. We detected theknocked-out protons inthe right HRSat two different kinemati-cal settings,
(θ
p,
pp)
= (48.
82◦, 1.481 GeV/c), and(58.
50◦, 1.246GeV/c), referred to here as “low pmiss” and “high pmiss”
respec-tively. These two settings cover a combined missingmomentum rangeof 40
≤
pmiss≤
550 MeV/c.Deuterium measurements wereonly done in the“low pmiss” kinematicsand thus extended only
uptopmiss
∼
300 MeV/c.Each HRS consisted of three quadrupole magnets for focus-ingandonedipolemagnetformomentumanalysis [43,44].These magnets were followed by a detector package, slightly updated with respect to the one in Ref. [43], consisting of a pair of vertical drift chambers used for tracking, and two scintillation counter planes that provide timing and trigger signals. A CO2
Cherenkov detector placed between the scintillators and a lead-glasscalorimeterplacedafterthemwereusedforparticle identifi-cation.
Electronswere selectedby requiringthat theparticle deposits morethanhalfofitsenergyinthecalorimeter: Ecal
| p|
>
0.
5.(
e,
ep)
coincidenceeventswereselectedbyplacinga
±
3σ
cutaroundthe relative electron andproton eventtimes.Due to the low experi-mentalluminosity,therandomcoincidenceeventratewas negligi-ble.We discardeda smallnumberofrunswithanomalous num-bersofeventsnormalizedtothebeamcharge.Measuredelectrons wererequiredtooriginate withinthe cen-tral
±
9 cm of the gas target to exclude events originating from thetargetwalls.Theelectronandprotonreconstructedtarget ver-tices were required to be within±
1.
2 cm of each other, which corresponds to±
3σ
of the vertex reconstruction resolution. By measuringscatteringfromanempty-cell-liketargetwedetermined thatthetargetcellwallcontributiontothemeasured(
e,
ep)
event yieldwasnegligible(1%).To avoid the acceptance edges of the spectrometer, we re-strictedtheanalysistoeventsthataredetectedwithin
±
4% ofthe centralspectrometermomentum,and±
27.
5 mrad inin-plane an-gle and±
55.
0 mrad in out-of-plane angle relative to the center ofthe spectrometeracceptance.In addition,we furtherrestricted the measurement phase-space by requiringθ
rq<
37.
5◦ tomini-mizethe effectofFSIand, inthehigh pmiss kinematics,xB
>
1.
3tofurthersuppressnon-QEevents.
The spectrometers were calibrated using sieve slit measure-mentstodefinescatteringanglesandbymeasuringthe kinemati-callyover-constrainedexclusiveH
(
e,
ep)
and2H(
e,
ep)
n reactions.The H
(
e,
ep)
reaction pmiss resolution was better than 9MeV/c.We verified the absoluteluminosity normalizationby comparing the measured elastic H
(
e,
e)
yield to a parametrization of the world data[45]. Wealso foundexcellent agreement betweenthe elasticH(
e,
ep)
andH(
e,
e)
rates,confirmingthatthecoincidence triggerperformedefficiently.Fig. 1 shows the number of measured 3H
(
e,
ep)
events asafunction of Emiss andof Q2 forthe low pmiss settingas well as
thesamedistributionscalculatedusingtheMonteCarlocodeSIMC [41] andnormalizedtogivethesameintegratednumberofevents asthedata.SIMCgenerated
(
e,
ep)
eventsusingEq. (3),withthe addition of radiation effects, that were then propagated through thespectrometer modelto accountforacceptance andresolution effects,andsubsequentlyanalyzedasthedata.The SIMC calcula-tionsuseda 3He spectralfunctioncalculatedby C.CiofidegliAtti4 Jefferson Lab Hall A Tritium Collaboration / Physics Letters B 797 (2019) 134890
Fig. 1. Number of3H(e,ep)events(counts)versusmissingenergyforthelow
p
misskinematics.Theblackmarkerscorrespondtothemeasureddata.Thelines corre-spondtothecalculateddistributionsobtainedfromaSIMC[41] simulationwitha spectralfunctioncalculatedbyC.CiofidegliAttiandL.P.Kaptari[42] and normal-izedtogivethesameintegralasthedata.Duetothelackof3H protonspectral functions,weassumedisospinsymmetryandusedthe3He neutronspectral func-tionforthe3H(e,ep)simulation (seetextfordetails).Theinsertshowsthe Q2 distributionforthesamekinematicalsetting.Seeonlinesupplementarymaterials forequivalent3He distributions.
andL.P.KaptariusingtheAV18potential [42].Duetothelackof
3H protonspectral functions, we assumedisospin symmetry and
used the 3He neutronspectral function for the 3H
(
e,
ep)
simu-lation.Thedifferencebetweenthecalculatedmomentum distribu-tionsofneutronsin3Heandprotonsin3H issmallandcontributes a3% uncertaintytothe3H
(
e,
ep)
calculationsandtothespectral-function ratio calculations [46]. The spectral function calculation appearstodescribethemeasured Q2 andE
missdistributionswell.
Seeonlinesupplementarymaterialsfordetailsandadditional com-parisons(including3He
(
e,
ep)
spectra).For each measured nucleus, we calculated the normalized
(
e,
ep)
eventyieldas:Y
(
pmiss)
=
N
(
pmiss)
C
·
tlive· (
ρ
/
A)
·
b,
(4)where A is the target atomic weight, N
(
pmiss)
is the number ofcountsforthat target in a givenbinof pmiss integrated overthe
experimental Emiss acceptance, C is the total accumulated beam
charge,tliveisthelivetimefractioninwhichthedetectorsareable
to collect data,
ρ
is the nominalareal density of the gasin the target cell,andb is a correction factorto account forchanges in thetargetdensitycausedbylocalbeamheating.b wasdetermined bymeasuringthebeamcurrentdependenceoftheinclusiveevent yield [40].Weformedthreeyieldratios,3He/d,3H/d,and3He/3H.We correctedthe measured ratioof thenormalized yields for the radioactive decayof 2
.
78±
0.
18% of the target 3H nucleito 3Heinthesixmonthssincethe targetwas filled,anddenotethecorrectedyieldratioby Rcorr.yield.
Thepoint-to-point systematical uncertaintieson thisratiodue to the event selection criteria (momentum and angular accep-tances, and
θ
rq and xB limits)were determined by repeating theanalysis5000times,selectingeachcriterionrandomlywithin rea-sonable limits foreach iteration. The systematicuncertainty was takentobethestandarddeviationoftheresultingdistributionof ratios.They rangefrom1% to 8% andare typically much smaller thanthestatisticaluncertainties.Thereisanoverallnormalization uncertaintyof 1
.
8%, predominantly duetothe target density un-certainty. Other normalization uncertainties due to beam-charge measurementandrun-by-runstabilityareatthe1% levelorlower, seeTable1.Seeonlinesupplementarymaterialsfordetails.Table 1
Systematicuncertaintiesintheextractionofthe3He/d,3H/d, and3He/3H(e,ep)normalizedevent-yieldratios,
R
corr.yield 3He/3H , (Fig. 2) and the 3He/3H cross-section ratio, σ3He(e,ep)/ σ3H(e,ep),(Fig.3).Uncertaintiesmarkedby‘*’contributeonly to thecross-sectionratio.Alluncertainties aresummedin quadrature.Seetextfordetails.
Overall Point-to-point
Targetwalls 1%
Targetdensity 1.5%
Beam-charge andstability 1%
Tritiumdecay 0.18% Cutsensitivity 1% - 8% Simulationcorrections* (bin-migration,radiation, Emacceptance) 1% - 2%
Fig. 2. Missingmomentumdependenceofthemeasured(e,ep)3He/d and3H/d
(top)and3He/3H (bottom) normalizedeventyieldratios.Thecirclesandsquares correspondrespectivelyto3He/d and3H/d inthetoppaneland tothelow and high
p
misssettingsinthebottompanel.Theerrorbarsincludebothstatisticalandpoint-to-pointsystematicaluncertainties.Anadditionaloverallnormalization uncer-taintyof1.8% isnotshown(seeTable1).ThesolidhistogramshowsthePWIASIMC simulationusingEq. (3) andthespectralfunctionofRef. [42] for
A
=3 andAV18 forA
=2.Thebinwidthsarethesameforthehistogramandthedata.Fig. 2 shows the missing momentum dependence of the cor-rected event yield ratios Rcorr3He/.yieldd , R
corr.yield
3H/d , and R
corr.yield
3He/3H for
each kinematical setting.The ratios of 3He and3H to deuterium arevery smallatlow pmiss,duetothemuchnarrowerdeuterium
momentumdistribution,andincreasetoaconstantvalueofabout two for 3H/d andabout three for3He/d at the largest measured
pmiss ofabout270 MeV/c.By contrast,the3He/3H ratio is about
threeatthesmallestmeasured pmiss anddecreasestoabout1.5at
pmiss
≈
250 MeV/c,withapossibleriseafterthat.Thisisconsistentwiththelow-pmissexpectation of2.5to3andslightlyhigherthan
theSRC-basedhigh-pmissexpectationofone.Thechangeinthe
ra-tiosismuchsmallerthanthefourorder-of-magnitudedecreasein thecalculatedmomentumdistributions(seeonlinesupplementary information).
Bothmeasured3He/d and3H/d ratiosareabout20% largerthan
thePWIAspectral-functionbasedSIMC calculation.This indicates that FSIeffects are the samefor both ratios.For thesame miss-ingmomentumrange,themeasured andcalculated3He/3H ratios
agree within the measurement accuracy of about 3%. This is a clearindication forcancellationof FSI effectinthe 3He/3H ratio. Athighermissing-momentum(pmiss
>
250 MeV/c),themeasured3He/3H ratiosareabout20
−
50% largerthanthecalculation.To extract the experimental cross-section ratio,
σ
3He(e,ep)/
σ
3H(e,ep)(
pmiss)
, we corrected the measured yield ratios usingSIMCforradiativeandbin-migrationeffectsaswellasforthefinite
Emiss acceptanceof thespectrometers. The finite Emiss correction
equalsthecalculatedmomentumdistributionratiodividedbythe calculatedratio of spectral functions integratedover the missing energy acceptance. The individual and total corrections were all lessthan 10% forall pmiss values.Weapply apoint-to-point
sys-tematicuncertaintyof20% oftheresultingcorrectionfactors.See Table1andonlinesupplementarymaterialfordetails.
We also calculated the final state interaction effects ofsingle rescatteringoftheknocked-outprotonwitheitherofthetwoother nucleons in the three-body-breakup reaction in the generalized Eikonalapproximation [47,48] using a computer code developed byM.Sargsian [49].ForeachbinwecalculatedboththePWIAand FSIcrosssectionandintegratedovertheexperimentalacceptance. FSIchangedtheindividual3He and3H
(
e,
ep)
cross-sectionsby be-tween10%and30%.However,theylargelycancelledinthedouble ratioRF S I
=
σ
F S I/
σ
P W I A|3Heσ
F S I/
σ
P W I A|
3H,
(5)producingatmosta5% effectatthehighestpmiss.Thisreinforces
the claim that FSI effects are very small in the cross-section ra-tio.WedidnotcorrectthedataforFSI.Seeonlinesupplementary materialsformoreinformation.
We tested the cross section factorization approximation by comparingthefactorizedspectralfunctionapproachusedinSIMC withanunfactorizedcalculationbyJ.Golak[50–52].Thedifference betweenthefactorizedandnon-factorizedcalculationswasabout 5%, which is not enough to explain the data-calculation discrep-ancyathighpmiss.
Fig. 3 shows the pmiss dependence of the extracted 3He/3H
(
e,
ep)
cross-sectionratio.Inthesimplestmodel,thisratioshould equaltwo,therelativenumberofprotonsin3He and3H.However, atlarge pmisstheratioshouldequalone,therelativenumberofnpSRCpairsin3He and3H [53–61].TheseSRCpairswillshiftequal
amountsofcross-section strengthfromlow pmiss tohighpmiss in
bothnuclei, increasing the3He to 3H ratio atlow pmiss tomore
thantwo.Themeasuredratiofollowsthissimplemodelofa tran-sition from independent nucleons at the lowest pmiss to np-SRC
pairs at higher pmiss, decreasing fromalmost three at low pmiss
towardsabout1.5atpmiss
=
250 MeV/c. Atlarger pmiss themea-suredratioisapproximatelyflat,withapossibleriseatthelargest
pmiss.
Withthemissing-energyacceptancecorrectionfor3He/3H and the small expected FSI effects, the resulting cross-section ratios should be sensitive to the ratio of momentum distributions. We
Fig. 3. The measured3He to3H cross-sectionratio,σ
3He(e,ep)/σ3H(e,ep)(pmiss),
plot-ted vs. pmiss compared with differentmodels ofthe corresponding momentum
distributionratio.Thefilledcircleandsquaremarkerscorrespondtothelowand high pmiss settingsrespectively. Uncertaintiesshowninclude bothstatisticaland
point-to-pointsystematicaluncertainties.Theoverallnormalizationuncertaintyof about1.8% isnotshown(seeTable1).Horizontalbarsindicatethebinsizesand areshownforonlythefirstandlastpointsineachkinematicalsettingasallother pointsareequallyspaced.Thebottompanelshowsthedoubleratioofdatato differ-entcalculatedmomentumdistributionratios,withthegreybandshowingthedata uncertainty.Thetheoreticalcalculationsaredoneusingdifferentlocalandnon-local interactions,aswellasdifferenttechniquesforsolvingthethree-bodyproblem.See textfordetails.
thereforecompare inFig. 3the measured cross-sectionratios di-rectly with the ratio of various single-nucleon momentum dis-tributions. The momentum distribution calculationsare obtained using either the variational Monte Carlo (VMC) technique with local interactions [46,62] or the Hyperspherical Harmonics (HH) method [63,64] withnon-localinteractions.
The local interactions used include the phenomenological AV18 [2] two-nucleon potential augmented by the Urbana X (UX) [65] three-nucleonforceandthechiralEFTpotentialsatN2LO (includingtwo- andthree-bodycontributions),usinga coordinate-spacecutoff of1 fm anddifferentparametrizations ofthe three-body contact term E
τ
and E1 [10,66–69]. Non-local interactions includethe meson-theoreticCD-Bonn [70] two-nucleonpotential, together withtheTucson-Melbourne [71] (TM) three-nucleon po-tential,orthelatestchiraltwo-bodypotentialsfromNLOtoN4LO [72], including three-nucleon interactions. The main contribution to the latter, namelythe one arising fromtwo-pion exchange, is effectively included atthe same chiral order asthe two-nucleon interaction,asexplainedinRefs. [64,72].Inthesecalculations, the momentum-spacecutoffiskeptfixedat500MeV.TheVMC cal-culations usingthe AV18andUXinteractions produce equivalent resultsastheHHcalculationsusingtheAV18plusUrbanaIX [73] interactions.
Forcompleteness,Fig.3alsoshowsthemomentum-distribution ratiocalculatedbyintegratingoverthemissingenergyinthe spec-tral functionsofRef. [42] and Ref. [74], obtainedusingthe AV18 two-nucleononlyandtheAV14 [75] two- andtheUrbanaVIII [76] (UVIII)three-nucleoninteractions,respectively.
Allcalculatedmomentum-distributionratiosshownagreewith the data up to pmiss
≈
250 MeV/c. At larger pmiss, thetheoreti-calpredictionsobtainedbyintegratingthespectralfunctionsorby calculatingthemomentum distributionratiowithlocalpotentials orwiththeCD-Bonn/TMmodeldisagreewiththedataby20–50%. In the caseof the non-local chiralpotential models,the calcula-tionsshowsignificantorderdependence.
Notethat,whilemomentumdistributionscalculatedwithlocal chiral-interactions dependstronglyonthecutoff parameter,these
6 Jefferson Lab Hall A Tritium Collaboration / Physics Letters B 797 (2019) 134890
effectsappeartomostlycancelintheratioofthemomentum dis-tributions [77].
Finally,although FSI calculated in the generalizedEikonal ap-proximation are small, more complete calculations are needed, including two- and three-body interaction operators [78], to de-termineifthediscrepancybetweendataandcalculationisdueto the reaction mechanismor to the validity of the underlying N N
potentialsatshort-distances. In addition,fullyrelativistic calcula-tionsareneededtoseeifthereareanysignificantcorrectionsdue tolongitudinal-transverseinterferenceeffects[30–32].
One possibleexplanation for thediscrepancy could be single-charge exchangeFSI,wherea struckneutronfroman SRC rescat-tersatalmost180◦ fromaproton,andtheprotonisdetected(np SCX), or a struck proton from an SRC rescatters at almost 180◦ froma neutron (pn SCX).A struckproton inan SRCrescattering from its partner neutron will decrease the number of observed proton events and a struck neutron in an SRC rescattering from its partner proton will increase the number of observed proton events. These two effects will largely cancel in both 3He
(
e,
ep)
and3H
(
e,
ep)
.However, in3He thestruckneutroninan SRCcan rescatter fromtheuncorrelatedproton, increasing thenumber of observedprotoneventsbutin3H it cannot.Thiscanincreasethe observed3He/3H ratio.Inaddition,iftheSCX occursatθ <
180◦, theneventsatsmallpmiss willbeobservedatlarger pmiss,ampli-fyingtheeffectsofSCXatlarge pmiss.
To summarize, we presented the first simultaneous measure-mentofthe3He
(
e,
ep)
,3H(
e,
ep)
andd(
e,
ep)
reactionsinkine-matics where the cross-sections are expected to be sensitive to theproton momentum distribution,i.e., atlarge Q2, xB
>
1, andθ
rq<
40◦ thatminimize two-bodycurrentsandtheeffects ofFSI.Wefurtherenhancedthesensitivitytothemomentumdistribution byextractingtheratioofthecross-sections,sothatmostofthe re-maining FSIeffects cancel,asconfirmedby ageneralizedEikonal approximationcalculationofleadingprotonrescattering.
Themeasured 3He/d and 3H/d correctedyield ratiosare small
atlow pmiss andincreasetothreeandtworespectivelyatpmiss
=
250 MeV/c.Both areabout20%lower thanPWIAcalculatedyield ratios,indicatingthatFSIeffectsareaboutthesameinbothpairs ofreactions.
While the measured corrected cross-section ratio
σ
3He(e,ep)/
σ
3H(e,ep)iswelldescribedbyPWIAcalculationsupto pmiss≈
250MeV/c, they disagree by only 20 - 50% at high pmiss, despite a
fourorder of magnitudedecrease of themomentum distribution inthisrange(seeFig. 2oftheonlinesupplementaryinformation). Thisisavastimprovementoverprevious
σ
3He(e,ep)measurementsatlower Q2 andx
B
=
1,whichdisagreedwithPWIAcalculationsbyfactorsofseveralatlarge pmiss [21,22]. This,together withFSI
calculations,stronglysupportsthereducedcontributionofnon-QE reactionmechanismsinourkinematics.
The data overall supports the transition from single-nucleon dominanceatlow pmiss,towards annp-SRCpairdominantregion
at high pmiss [53–61]. However, more complete calculations are
neededtoassess theimplications ofthe observed20–50% devia-tionofthedatafromthePWIAcalculationintheexpectednp-SRC
pair dominance region,including the effectsof single charge ex-change.Iftheobserveddifferencebetweenthe3He/3H
experimen-tal ratioandmomentum distribution ratios atlarge missing mo-mentaisduetotheunderlyingN N interaction,thenitcanprovide significant new constraints on the previously loosely-constrained short-distancepartsofthe N N interaction.
Acknowledgements
We acknowledge the contribution of the Jefferson-Lab target group and technicalstaff for design andconstruction ofthe
Tri-tium targetandtheir supportrunningthisexperiment.Wethank C. Ciofi degli Atti and L. Kaptari for the 3He spectral function calculations andM. Sargsian,M. Strikman, J. Carlson, S. Gandolfi, and R. B. Wiringa for manyvaluable discussions. This work was supported by the U.S. Department of Energy (DOE) grant DE-AC05-06OR23177 under which Jefferson Science Associates, LLC, operatestheThomasJeffersonNationalAcceleratorFacility,theU.S. National Science Foundation, the Pazi foundation, the Israel Sci-enceFoundation,andtheNUCLEISciDAC program.Computational resourcesforthecalculationoftheN2LOmomentumdistributions havebeen providedby LosAlamosOpen Supercomputingvia the Institutional Computing(IC)program andby theNational Energy ResearchScientificComputingCenter(NERSC),whichissupported bytheU.S.DepartmentofEnergy,OfficeofScience,underContract No.DE-AC02-05CH11231.TheKentStateUniversitycontributionis supported under the PHY-1714809 grant from the U.S. National Science Foundation. The University of Tennessee contribution is supported by the DE-SC0013615 grant. The work of ANL group membersissupportedbyDOEgrantDE-AC02-06CH11357. Appendix A. Supplementarymaterial
Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi.org/10.1016/j.physletb.2019.134890.
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